'''Definition''': Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes.

'''Definition''': Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes.

-

<br />'''Probability density function''': For <math>X\sim Pareto(x_m,\alpha)\!</math>, the Pareto probability density function is given by

+

<br />'''Probability density function''': For <math>X\sim \operatorname{Pareto}(x_m,\alpha)\!</math>, the Pareto probability density function is given by

*The severity of large casualty losses for certain businesses, such as general liability, commercial auto, and workers compensation

*The severity of large casualty losses for certain businesses, such as general liability, commercial auto, and workers compensation

-

===Example===

===Example===

-

Suppose that the income of a certain population has a Pareto distribution with <math>\alpha=3</math> and <math>x_m=1000</math>. Compute the proportion of the population with incomes between 2000 and 4000.

+

Suppose that the income of a certain population has a Pareto distribution with <font size="3"><math>\alpha=3</math></font> and <font size="3"><math>x_m=1000</math></font>. Compute the proportion of the population with incomes between 2000 and 4000.

We can compute this as follows:

We can compute this as follows:

Line 59:

Line 60:

The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html SOCR distributions]

The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html SOCR distributions]

Contents

Pareto Distribution

Definition: Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes.

Probability density function: For , the Pareto probability density function is given by

where

xm is the minimum possible value of X

α is a positive parameter which determines the concentration of data towards the mode

x is a random variable (x > xm)

Cumulative density function: The Pareto cumulative distribution function is given by

where

xm is the minimum possible value of X

α is a positive parameter which determines the concentration of data towards the mode

Applications

The Pareto distribution is sometimes expressed more simply as the “80-20 rule”, which describes a range of situations. In customer support, it means that 80% of problems come from 20% of customers. In economics, it means 80% of the wealth is controlled by 20% of the population. Examples of events that may be modeled by Pareto distribution include: